We continued by introducing a bundle for the eigenstates. A remarkable property of this bundle is that it is not a principal bundle, but has the structure of a semi-principal bundle, which is described in [1]}. That is, each of its fibers consists of a collection of eigenrays rather than a single eigenray. We found a natural connection on this bundle, which provides a parallel transport description of the adiabatic geometric phase as found by Garrison & Wright . Because of the semi-principal structure, this incorporates non-cyclic states as well. This formalism hence extends previous holonomy formulations, which are based on principal bundles. Among these are the first holonomy interpretation reported by Simon , which treats the Hermitian case with separate energy bands, and the formalism in , which considered the non-Hermitian case restricted to cyclic states. We furthermore found that the Aharonov-Anandan formalism [2]} for general Hermitian systems can be obtained by a reduction from the present formalism using a given inner product. Moreover, we found that the dynamical phase can be included as a non-geometric addition, which means that the full adiabatic evolution of a state can be calculated naturally on this bundle. We then obtained a generalized quantum geometric tensor, which we found to be directly related to covariant derivatives.
